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Q5:
How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which are divisible by 10 and no digit is repeated ?
Solution :
Step 1: Identifying the Constraints
We are tasked with forming a 6-digit number using the set of digits $S = \{0, 1, 3, 5, 7, 9\}$ such that:
- The number must be divisible by $10$. [A number is divisible by $10$ if and only if its units digit is $0$].
- No digit is repeated. [Since there are exactly 6 unique digits provided and we must form a 6-digit number, we must use each digit exactly once].
Step 2: Defining the Positional Requirements
A 6-digit number has six positions: $P_6 P_5 P_4 P_3 P_2 P_1$.
| Position | Constraint |
|---|---|
| $P_1$ (Units Place) | Must be $0$ for divisibility by $10$. |
| $P_6, P_5, P_4, P_3, P_2$ | Remaining digits $\{1, 3, 5, 7, 9\}$ must occupy these positions. |
Step 3: Calculating Permutations for Each Position
- Fixing the Units Place ($P_1$): There is only $1$ way to place the digit $0$. [Constraint: Divisibility by $10$ rule].
- Filling the Remaining Positions ($P_6$ to $P_2$): We have $5$ remaining digits $\{1, 3, 5, 7, 9\}$ to be arranged in the remaining $5$ positions. The number of ways to arrange $n$ distinct objects in $n$ positions is given by $n!$ [By Fundamental Counting Principle].
Step 4: Algebraic Derivation
Total numbers $N$ is calculated as:
$N = (\text{Choices for } P_1) \times (\text{Permutations of remaining } 5 \text{ digits in } 5 \text{ positions})$
$N = 1 \times 5!$
$N = 1 \times (5 \times 4 \times 3 \times 2 \times 1)$
$N = 120$
Step 5: Verification of Digit Constraints
Since the set $S$ contains $0$, we must ensure $0$ is not in the $P_6$ (leading) position. However, since $P_1$ is fixed as $0$, the digit $0$ cannot occupy $P_6$. Thus, all $120$ permutations satisfy the criteria for a valid 6-digit number.
Final Solution: 120
More Questions from Class 11 Mathematics Permutations and Combinations Miscellaneous Exercise on Chapter 6
- Q1: How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER ?
- Q2: How many words, with or without meaning, can be formed using all the letters of the word EQUATION at a time so that the vowels and consonants occur together?
- Q3(i): A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: (i) exactly 3 girls ?
- Q3(ii): A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: (ii) atleast 3 girls ?
- Q3(iii): A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: (iii) atmost 3 girls ?
- Q4: If the different permutations of all the letter of the word EXAMINATION are listed as in a dictionary, how many words are there in this list before the first word starting with E ?
- Q6: The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet ?
- Q7: In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions ?
- Q8: Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.
- Q9: It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible ?
- Q10: From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen ?
- Q11: In how many ways can the letters of the word ASSASSINATION be arranged so that all the $S$'s are together ?
CBSE Solutions for Class 11 Mathematics Permutations and Combinations
Chapters in CBSE - Class 11 Mathematics
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